Executive Overview xxiii

0 My First Monte Carlo Application One-Factor Problems 1

0.1 Introduction and objectives 1

0.2 Description of the problem 1

0.3 Ordinary differential equations (ODE) 2

0.4 Stochastic differential equations (SDE) and their solution 3

0.5 Generating uniform and normal random numbers 4

0.6 The Monte Carlo method 8

0.7 Calculating sensitivities 9

0.8 The initial C++ Monte Carlo framework: hierarchy and paths 10

0.9 The initial C++ Monte Carlo framework: calculating option price 19

0.10 The predictor-corrector method: a scheme for all seasons? 23

0.11 The Monte Carlo approach: caveats and nasty surprises 24

0.12 Summary and conclusions 25

0.13 Exercises and projects 25

PART I FUNDAMENTALS

1 Mathematical Preparations for the Monte Carlo Method 31

1.1 Introduction and objectives 31

1.2 Random variables 31

1.3 Discrete and continuous random variables 34

1.4 Multiple random variables 37

1.5 A short history of integration 38

1.6 ¿-algebras, measurable spaces and measurable functions 39

1.7 Probability spaces and stochastic processes 40

1.8 The Ito stochastic integral 41

1.9 Applications of the Lebesgue theory 43

1.10 Some useful inequalities 45

1.11 Some special functions 46

1.12 Convergence of function sequences 48

1.13 Applications to stochastic analysis 49

1.14 Summary and conclusions 50

1.15 Exercises and projects 50

2 The Mathematics of Stochastic Differential Equations (SDE) 53

2.1 Introduction and objectives 53

2.2 A survey of the literature 53

2.3 Mathematical foundations for SDEs 55

2.4 Motivating random (stochastic) processes 59

2.5 An introduction to one-dimensional random processes 59

2.6 Stochastic differential equations in Banach spaces: prologue 62

2.7 Classes of SIEs and properties of their solutions 62

2.8 Existence and uniqueness results 63

2.9 A special SDE: the Ito equation 64

2.10 Numerical approximation of SIEs 66

2.11 Transforming an SDE: the Ito formula 68

2.12 Summary and conclusions 69

2.13 Appendix: proof of the Banach fixed-point theorem and some applications 69

2.14 Exercises and projects 71

3 Alternative SDEs and Toolkit Functionality 73

3.1 Introduction and objectives 73

3.2 Bessel processes 73

3.3 Random variate generation 74

3.4 The exponential distribution 74

3.5 The beta and gamma distributions 75

3.6 The chi-squared, Student and other distributions 79

3.7 Discrete variate generation 79

3.8 The Fokker-Planck equation 80

3.9 The relationship with PDEs 81

3.10 Alternative stochastic processes 84

3.11 Using associative arrays and matrices to model lookup tables and volatility surfaces 93

3.12 Summary and conclusion 96

3.13 Appendix: statistical distributions and special functions in the Boost library 97

3.14 Exercises and projects 102

4 An Introduction to the Finite Difference Method for SDE 107

4.1 Introduction and objectives 107

4.2 An introduction to discrete time simulation, motivation and notation 107

4.3 Foundations of discrete time approximation: ordinary differential equations 109

4.4 Foundations of discrete time approximation: stochastic differential equations 113

4.5 Some common schemes for one-factor SDEs 117

4.6 The Milstein schemes 117

4.7 Predictor-corrector methods 118

4.8 Stiff ordinary and stochastic differential equations 119

4.9 Software design and C++ implementation issues 125

4.10 Computational results 126

4.11 Aside: the characteristic equation of a difference scheme 127

4.12 Summary and conclusions 128

4.13 Exercises and projects 128

5 Design and Implementation of Finite Difference Schemes in Computational Finance 137

5.1 Introduction and objectives 137

5.2 Modelling SDEs and FDM in C++ 137

5.3 Mathematical and numerical tools 138

5.4 The Karhunen-Loeve expansion 143

5.5 Cholesky decomposition 144

5.6 Spread options with stochastic volatility 146

5.7 The Heston stochastic volatility model 153

5.8 Path-dependent options and the Monte Carlo method 160

5.9 A small software framework for pricing options 161

5.10 Summary and conclusions 162

5.11 Exercises and projects 162

6 Advanced Finance Models and Numerical Methods 167

6.1 Introduction and objectives 167

6.2 Processes with jumps 168

6.3 Levy processes 171

6.4 Measuring the order of convergence 172

6.5 Mollifiers, bump functions and function regularisation 176

6.6 When Monte Carlo does not work: counterexamples 177

6.7 Approximating SDEs using strong Taylor, explicit and implicit schemes 179

6.8 Summary and conclusions 183

6.9 Exercises and projects 184

7 Foundations of the Monte Carlo Method 189

7.1 Introduction and objectives 189

7.2 Basic probability 189

7.3 The Law of Large Numbers 190

7.4 The Central Limit Theorem 191

7.5 Quasi Monte Carlo methods 194

7.6 Summary and conclusions 198

7.7 Exercises and projects 198

PART II DESIGN PATTERNS

8 Architectures and Frameworks for Monte Carlo Methods: Overview 203

8.1 Goals of Part II of this book 203

8.2 Introduction and objectives of this chapter 203

8.3 The advantages of domain architectures 204

8.4 Software Architectures for the Monte Carlo method 207

8.5 Summary and conclusions 212

8.6 Exercises and projects 213

9 System Decomposition and System Patterns 217

9.1 Introduction and objectives 217

9.2 Software development process; a critique 217

9.3 System decomposition, from concept to code 217

9.4 Decomposition techniques, the process 220

9.5 Whole-part 222

9.6 Whole-part decomposition; the process 223

9.7 Presentation-Abstraction Control (PAC) 226

9.8 Building complex objects and configuring applications 229

9.9 Summary and conclusions 239

9.10 Exercises and projects 239

10 Detailed Design using the GOF Patterns 243

10.1 Introduction and objectives 243

10.2 Discovering which patterns to use 244

10.3 An overview of the GOF patterns 255

10.4 The essential structural patterns 257

10.5 The essential creational patterns 266

10.6 The essential behavioural patterns 270

10.7 Summary and conclusions 276

10.8 Exercises and projects 276

11 Combining Object-Oriented and Generic Programming Models 281

11.1 Introduction and objectives 281

11.2 Using templates to implement components: overview 281

11.3 Templates versus inheritance, run-time versus compile-time 283

11.4 Advanced C++ templates 286

11.5 Traits and policy-based design 294

11.6 Creating templated design patterns 306

11.7 A generic Singleton pattern 307

11.8 Generic composite structures 310

11.9 Summary and conclusions 314

11.10 Exercises and projects 314

12 Data Structures and their Application to the Monte Carlo Method 319

12.1 Introduction and objectives 319

12.2 Arrays, vectors and matrices 319

12.3 Compile-time vectors and matrices 324

12.4 Creating adapters for STL containers 331

12.5 Date and time classes 334

12.6 The class string 339

12.7 Modifying strings 343

12.8 A final look at the generic composite 345

12.9 Summary and conclusions 348

12.10 Exercises and projects 348

13 The Boost Library: An Introduction 353

13.1 Introduction and objectives 353

13.2 A taxonomy of C++ pointer types 353

13.3 Modelling homogeneous and heterogeneous data in Boost 361

13.4 Boost signals: notification and data synchronisation 367

13.5 Input and output 368

13.6 Linear algebra and uBLAS 371

13.7 Date and time 372

13.8 Other libraries 372

13.9 Summary and conclusions 374

13.10 Exercises and projects 374

PART III ADVANCED APPLICATIONS

14 Instruments and Payoffs 379

14.1 Introduction and objectives 379

14.2 Creating a C++ instrument hierarchy 379

14.3 Modelling payoffs in C++ 383

14.4 Summary and conclusions 392

14.5 Exercises and projects 393

15 Path-Dependent Options 395

15.1 Introduction and objectives 395

15.2 Monte Carlo - a simple general-purpose version 396

15.3 Asian options 401

15.4 Options on the running Max/Min 411

15.5 Barrier options 412

15.6 Lookback options 418

15.7 Cliquet Options 422

15.8 Summary and conclusions 424

15.9 Exercises and projects 424

16 Affine Stochastic Volatility Models 427

16.1 Introduction and objectives 427

16.2 The volatility skew/smile 427

16.3 The Heston model 429

16.4 The Bates/SVJ model 441

16.5 Implementing the Bates model 443

16.6 Numerical results - European options 444

16.7 Numerical results - skew-dependent options 446

16.8 XLL - using DLL within Microsoft Excel 449

16.9 Analytic solutions for affine stochastic volatility models 455

16.10 Summary and conclusions 457

16.11 Exercises and projects 458

17 Multi-Asset Options 461

17.1 Introduction and objectives 461

17.2 Modelling in multiple dimensions 461